For this analysis, we considered answers between 800 and 1015 to be reasonably accurate, as was a clear statement that the answer was 98 times 9. This level of accuracy is different from that credited by NEMP, which limited the range of credited answers to those between 850 and 900. We chose a somewhat more lax range in analysing language, as it gave us more students whose answers were likely to include mathematical language. The major difference in success on this item, by our criteria, was between Year 4 and Year 8 students.

Of the Year 4 students, 9 gave answers that were within this range. Correct answers came from each group analysed. Three of the six high decile girls gave approximately correct answers, while in other groups one or two students were relatively accurate. Half of the appropriate answers came from the high decile students.

A total of 29 Year 8 students gave answers in this range or gave the calculation as 98 times 9. Again, some correct answers were given by students in each subgroup, between 3 correct answers (low decile non-Pacific boys) and 6 correct answers (high decile girls). There was no overall pattern for success by gender, ethnicity or economic background in the proportion of students who were reasonably correct. See Table 5.1.

The answers that students gave could be classified as numerical, mathematical explanation, non-mathematical stories, and statements that indicated not knowing or uncertainty (e.g. 'I guessed', 'don't know').

All but one student gave a numerical answer. For Year 4 students, these answers ranged between 90 and 1 000 000. For the Year 8 students, the answers ranged between 108 and 48000. Two Year 8 student said that the answer would be ninetyeight times nine and did not estimate what 98 x 9 would equal.

Both low and high decile year 4 students gave stories without mathematical content. The nature of these stories is covered more fully in Chapter 7. The stories of Year 4 students included:

I got no idea. Oh, because they busy? And they waiting for the stop at traffic light. [Q] Because there's many cars, there's heaps. [Q] And they're waiting at the traffic not working, won't work, and that's all (Year 4 Pacific low decile boy)

Because the motorway's kind of busy and people don't really umm hold up on the motorway, because you can kind of, because there basically isn't a speed limit on the motorway, they go quite fast. (High decile Year 4 girl)

The two Year 8 non-mathematical stories could be considered protoquantitative, using the terms 'too many' and 'lots of' to describe the reason why there would be many cars.

because there's too many cars. [Q] there's too many cars in the traffic (Year 8 low decile non-Pacific girl)

None of stories told by the Year 4 students had a numerical answer that was approximately right. For the three stories given above, the numerical answers were 89, 52 and a 1000 respectively. The Year 8 student had an approximation of 1000, which could indicate some number sense in the absence of the ability to explain.

In addition to these explanatory stories, some Year 4 students and some interviewers personalised the story. These examples were conversational. For example, one interviewer said 'That's a lot more cars than go by here, isn't it'. A Pacific Year 4 girl asked, 'Is this New Zealand?' Another conversational example came from a high decile boy who said, 'if it was Auckland, it would be a typical day', and then moved to working with decontextualised numbers. This discussion about the picture did not occur with Year 8 students, and may have been an attempt to keep rapport going with the younger students.

For the Year 4 students, these stories suggest that, in the absence of a ready mathematical explanation, a story about the context seemed the best answer to these students. As with the stories told by different economic groups in response to the Bank Account task, the stories were related to their personal experience.

How one does a mathematical operation mentally is not easy to explain, as found in experience with the Numeracy Project (2004). Explanations considered mathematical among those given by these students ranged between protoquantitative ones that did not include numbers and said 'many more' or 'heaps…more', and those that mentioned an operation like 'count them' to those that explained the multiplication followed by repeated subtraction of 2. Examples of these are given in Table 5.3.

Of the nine reasonably correct answers from Year 4 students, four were accompanied by explanations that could be followed. It is possible that the other reasonable answers were guesses that could not be explained or that the student found them in an appropriate manner but then could not explain what they had done. All of the Year 8 accurate answers were easy to follow and usually concise.

The relatively accurate answers mostly describe a combination of multiplication, addition and subtraction. Given the fact that these students were describing a mental function that was sometimes lengthy, some of these explanations were remarkably good. Even conceptually clear answers included restarted phrases (I did nine, ninety times nine) and a second start at the sentence as below.

Well, I did nine, ninety nine times nine, I thought well I'm not going into ninety times nine so I just did ninety times ten is nine hundred so. (Year 4 high decile boy)

Note that in this section the boy uses 'did' as the verb to describe his mathematical manipulation rather than 'multiplied', and a colloquial 'I'm not going in to' to describe a choice abandoned.

Another well-described procedure was:

I just um, went up to 98 plus two is a hundred and nine a hundreds are nine hundred take away.. two nines, oh, um… [Q] Um, I forgot how I done that .. um, it was... [Q] Ninety eight, plus two is a hundred and nine a hundreds are nine hundred take away two, and then, um, take away two nines and I got eight hundred and eighty eight. (Year 4 low decile non-Pacific boy)

The only absolutely accurate answer from a Year 4 student started by describing one method for multiplication, repeated doubling, and then moved to repeated subtraction. This boy thought out loud showing how he found his answer before giving being asked for an explanation. His explanation includes self-corrections and shaking his head when he decided that his way of starting was not going to be useful. ...

(silent, thinking, then counts very quietly to himself)...which makes a hundred and eighty, six, another one that makes a hundred and eighty .. I mean a hundred and ninety, six, and a hundred and ninety four .. wait (shaking his head)...nine hundred, nine hundred and eighty, nine hundred and ninety two .. eight hundred and ninety .. ninety eight, eight hundred and ninety six, eight hundred and ninety four, eight hundred and ninety two, eight hundred and ninety, eight hundred and eighty eight, eight hundred and eighty six, eight hundred and eighty four, eight hundred and eighty two, eight hundred and eighty two.
Q
Um well, what I did is I added up, um, I went from nine, nine hundred from ninety eight and I made that a hundred then I subtracted nine, nine twos from it, from the answer.

In this protocol 'makes' is used for 'equals'.

Other students who gave accurate answers gave considerably less clear explanations, possibly because they were not able to recall how they did the problem. For example, a girl who gave her answer as 998 started using repeated addition in her explanation and then gave up. We are left uncertain about how she got her answer, which could have been done by multiplying 9 x 10 and adding 98.

Umm .. ninety eight, plus ninety eight, plus ninety eight, plus ninety eight, plus ninety eight, plus ninety eight, plus ninety eight, no, I can't do it. (Year 4 high decile girl)

Another student who gave 1000 as his answer said:

Because um, ninety and ninety makes a hundred and something .. an then there'll just be ten hundred.' (Year 4 high decile boy)

Incorrect answers sometimes included a recognisable procedure. One student described the way in which nine times tables could be worked out on your fingers or knuckles. Another student described the vertical algorithm, but added the products ignoring the fact that the first amount was not 9 times 9 but 90 times 9

. …it's fifty, a hundred and um fifty three.
Q
..um, because um, nine times nine is eighty one, and the ninth um eight times nine is seventy two, and I put it together. (Year 4 low decile non-Pacific boy)

The most common answers from Year 8 students were versions of ninety-eight times nine or 100 times 9. These students did not appear to have as much difficulty in explaining the mathematical path to their solution as the Year 4 students did.

This question aroused less discussion than did the other items. As a quick index of the amount of talk from each ethnic and economic group, the number of linguistic elements recorded was totalled. For high decile students, 270 elements were recorded; for low decile non-Pacific students, 184 elements were recorded; and for low decile Pacific students, 100 elements were recorded. Although rough, this is one index of the willingness of these students to discuss mathematics with the interviewers. It is compatible with the argument that found that students are more likely to carry on conversations with people from similar backgrounds than with people of different backgrounds (Bogdan & Biklen, 1982).

In the following section, totals usually equal more than 100% as students used more than one type of each element in their answers.

The form of the second question tells the student to: 'Explain to me how you got your answer'. Thus it would be conversationally appropriate for a student to use a personal pronoun in their answer.

Nancy Barron (1971) found that men and women teachers chose to use different grammatical cases in their classroom discussions. She hypothesised that this reflected differences in what was valued by the different gender groups.

Women produced a significantly greater proportion of explicit participative cases than men, thus demonstrating their greater concern with internal physiological states. The greater involvement of men with implementation of action by means of objects was shown by their greater use of instrumental or source cases. (p. 39)

As a result of this, it seemed interesting to code who or what was the main actor in the majority of clauses of each response. For this task, the actors were either 'I', a generic 'you' or an impersonal actor such as a number or calculation or the question itself. These final type of impersonal actors are discussed further in the section on hesitant language. The distribution of these is given in the following table.

Certainly the gender difference that Barron had found with teachers was not found with students when responding to this task.

Sixty-one percent of the Year 4 students and 81% of the Year 8 students used 'I', 'you', or an implied personal pronoun as agents in their explanations. Among Year 4 students, Pacific students were more likely to use either 'it' or an existential agent (e.g. 'there are'). This difference is significant (Fisher's exact test 2-tailed p=0.0031). For Year 8 students there was almost no difference amongst groups.

Twenty-two, or 60% of Year 8 students used numbers as the agent of their sentences or clauses, as in 'because 98 plus 2…', while only six (17% of the Year 4 students) limited their explanations to having the numbers as the agent. For both year levels, this use was spread across the groups. The difference appeared to be related to competence in the Year 8 students' ability to strip away the story shell and deal with the mathematics only, while the younger students talked about what they did or what the cars did.

This was the one task of the four considered in this study for which mathematical verbs predominated for both year levels. These verbs included 'times', 'take away', 'equals' and 'estimate'. One third (33%) of all verbs by Year 4 students and nearly half (48%) of those used by Year 8 students used were considered mathematical. The second largest category was static verbs.

In Year 4, this preponderance of mathematical verbs over all other types of verbs is mostly due to the upper decile students who said more and gave more mathematical explanations (Fisher's 2-tailed test p = 0.0472).

In Year 8, the highest proportion of mathematical verbs was used by low decile non- Pacific boys but this proportion was influenced by a high number used by two of the boys who gave extensive answers. See Table 5.5.

There were 18 responses which used estimations to gain either an approximate or exact amount. Over half of these were by students attending high decile schools, twice as many students as those from low decile schools, either Pacific Islanders or not. There were equivalent numbers of boys and girls, but more than three quarters were students in Year 8. Similar results for year level and gender can be seen for the 14 responses which gave '9 x 98' for their calculation. However, there were no significant differences in the decile level of schools attended. At the other end of the table, of the 20 students who were unable to use information mathematically, 7 students were from high decile schools. Six students were from low decile schools in Pacific communities. There was a similar even distribution by gender. One difference in distribution was that 16 of these 20 students were in Year 4. This suggests that students from high decile schools were most likely to use estimates whilst non-mathematical responses were more evenly spread across groups.

Of 26 students who used clear language, 20 were in Year 8, suggesting that clear language is related to age regardless of the mathematical ability of the student. Students were clearer in Year 8 whether they gave an appropriate mathematical response or not. There were no other differences according to gender, decile level of school attended or ethnicity. There were only 8 students who gave elliptical responses with no clear distinctions between any group.

The Motorway task required students to provide an explanation of their calculation to determine the number of cars. As such, it shared many similarities with the Weigh Up task which also required an explanation in its Plan and Explanation sections. It was, therefore, not surprising to find many of the same text elements being used; that is Premise, Consequences, Conclusions and Elaborators. The distribution of students using these elements is given in Table 5.7.

More students attending high decile schools used combinations of three or more text elements. As was seen in Table 5.6, Pacific students predominantly only used a Premise. The next few tables go through each of the text elements and provide more details before looking at what combinations of text elements were most likely to be related to clear language and accurate responses.

Introductions only occurred at the beginning of the response and seemed to personalise it. For example, one boy stated

Um, I go for, I go to nine times nine then eight times nine, and then, then I, then I plussed the two answers together and got around, that.

'I go to' is setting the scene in that he is describing what he did before going on to describe the actual process.

As seen in Table 5.9, eleven students began their responses with an Introduction. All were followed by a Premise and two continued with a Consequence, while another six used a Premise, a Consequence and a Conclusion. It was, therefore, more likely that an Introduction was found in texts which incorporated a longer, more complex series of elements.

In Table 5.10 there are no clear distinctions in who used Conclusions on the basis of age, although there does seem to be a slight tendency for boys rather than girls to use them. No Conclusions were found at the beginning of text structures, unlike what had occurred in the Better Buy responses. It was far more likely that students from high decile schools included them in their responses. Unlike the Better Buy texts, only two of these Conclusions were implicit. One other small difference is that Year 8 students were more likely to have a Conclusion preceded by a Consequence, whereas a Conclusion preceded only by a Premise was more likely to be used by a Year 4 student. However, the numbers are small and any pattern remains uncertain.

Table 5.11 shows the distribution of students who concluded their responses with a Consequence. As there were differences between Premises, the type of preceding Premise is also provided. Premises are divided into types based on the kind of evidence they used. A factual Premise used information given in the question or from the photo which was supplied, for example 'because it says nine minutes', 'because there's lots of cars going up and down'. A mathematical Premise used a mathematical calculation or made reference to a mathematical fact, such as 'I rounded ninety-eight up to, umm, one hundred' or 'ninety and ninety makes a hundred and something'. When students used more than one Premise, they were categorised under the first Premise or the type of Premise used most often.

Of the 24 Pacific students, twenty used only Premises. Of these, twelve were either factual, repeating information in the question, or personal. Personal Premises were related to an action that the speaker did which was not mathematical, for example 'I don't know I just guessed'. Personal Premises, because of their nature, did not occur with a Consequence and/or Conclusion. It would seem that more Pacific students were likely to feel that admitting a lack of knowledge was an acceptable way to respond to a mathematical assessment task. However, it was also the case that 8 Pacific students recognised that mathematical calculation was required and were able to provide an appropriate one. For example a Year 8 girl said, 'Nine minutes times ninety-eight cars'. Yet, knowing this did not mean that they could actually provide an appropriate numerical amount, as many found doing this calculation too hard to do mentally and had no other way of arriving at an approximation. 4 Pacific Year 8 students did the calculation and gave an accurate response by just supplying a Premise. However, if only a mathematical Premise was given then it was unlikely that the student would be able to solve the problem appropriately. The number of Pacific students who only used a Premise was double the number of students from low decile schools in areas with a low Pacific population.

In considering how the different elements were connected, there were not the number of Premises beginning with 'if … ' as there had been in the Better Buy task. This can be seen in Table 5.13.